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Next: The Maximum Data Rate Up: ELEC350 Communications Theory and Previous: ELEC350 Communications Theory and

Derivation the dB version of the Path Loss Equation for Free Space.

For free propagation waves in radio channel, the path loss model is


\begin{displaymath}L_0 = { \left ( \frac{4 ~ \pi ~ d}{\lambda} \right ) }^2
\end{displaymath} (1)

Where $\lambda = c / f_c $ , so


\begin{displaymath}L_0 = { \left ( \frac{4 ~ \pi ~ d ~ f_{c}}{c} \right ) }^2
\end{displaymath} (2)

For d in meters , fc in GHz and $ c = 3 \times 10^8 $ meters / second,


\begin{displaymath}L_0 = {\left ( \frac{4 ~ \pi ~ d ~ f_c \times 1 \times 10^9}{3 \times 10^8} \right ) }^2
\end{displaymath} (3)

By taking $ \log_{10} \left ( \log \right ) $ of both sides of equation $ \left ( 3 \right ) $ to obtain the dB version


L0 = $\displaystyle 10 \log { \left ( \frac{4 ~ \pi ~ d ~ f_c \times 1 \times 10^9 }{3 \times 10^8 } \right ) }^2$  
  = $\displaystyle 20 \log \left ( \frac { 4 \pi \times 10 }{3} \right ) + 20 \log \left ({f_c} \right ) + 20 \log \left ({d} \right )$  
  = $\displaystyle 32.4 + 20 \log \left ({f_c} \right ) + 20 \log \left ({d} \right )$ (4)


next up previous
Next: The Maximum Data Rate Up: ELEC350 Communications Theory and Previous: ELEC350 Communications Theory and
Yajun Kou
2000-09-19